Assessment of heterogeneity of residual variances using changepoint techniques

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Abstract : Several studies using test-day models show clear heterogeneity of residual variance along lactation. A changepoint technique to account for this heterogeneity is proposed. The data set included 100 744 test-day records of 10 869 Holstein-Friesian cows from northern Spain. A three-stage hierarchical model using the Wood lactation function was employed. Two unknown changepoints at times $T {1}$ and $T {2}$, $0 < T {1} < T {2} < t {\max}$, with continuity of residual variance at these points, were assumed. Also, a nonlinear relationship between residual variance and the number of days of milking $t$ was postulated. The residual variance at a time $t$ $\sigma {et}^2$ in the lactation phase $i$ was modeled as: $\sigma {et}^2 = t^{\lambda i} \sigma {ei}^2$ for $i =1,2,3$, where $\lambda i$ is a phase-specific parameter. A Bayesian analysis using Gibbs sampling and the Metropolis-Hastings algorithm for marginalization was implemented. After a burn-in of 20 000 iterations, 40 000 samples were drawn to estimate posterior features. The posterior modes of $T {1}$, $T {2}$, $\lambda {1}$, $\lambda {2}$, $\lambda {3}$, $\sigma {e1}^{2}$, $\sigma {e2}^{2}$, $\sigma {e3}^{2}$ were $53.2$ and $248.2$ days; $0.575$, $-0.406$, $0.797$ and $0.702$, $34.63$ and $0.0455 { m kg}^{2}$, respectively. The residual variance predicted using these point estimates were $2.64$, $6.88$, $3.59$ and $4.35 { m kg}^{2}$ at days of milking $10$, $53$, $248$ and $305$, respectively. This technique requires less restrictive assumptions and the model has fewer parameters than other methods proposed to account for the heterogeneity of residual variance during lactation.

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